In an election between two candidates, 85% of the registered voters cast their vote and 4% of the votes polled were found invalid. The winning candidate got 65% of the valid votes and won the election by a margin of 2142 votes. How many voters were registered?

Correct Answer: The correct option is 8750 (Option 2).

Step-by-Step Explanation:

Step 1: Define the variable
Let the total number of registered voters be $x$.

Step 2: Calculate the number of votes polled
According to the problem, 85% of the registered voters cast their vote.
So, the number of votes polled is:

$\text{Votes polled}=0.85x$

Step 3: Calculate the number of valid votes
Since 4% of the votes polled were invalid, 96% of the polled votes were valid (100% - 4% = 96%).
So, the number of valid votes is:

$\text{Valid votes}=96\%\text{ of }0.85x$

$\text{Valid votes}=0.96\times 0.85x=0.816x$

Step 4: Determine the winning margin in terms of valid votes
The winning candidate got 65% of the valid votes.
Therefore, the losing candidate received 35% of the valid votes (100% - 65% = 35%).
The difference (margin) between the winner and the loser is:

$\text{Margin percentage}=65\%-35\%=30\%\text{ of valid votes}$

Step 5: Set up the equation and solve for $x$
We are given that the winning margin is 2142 votes. So:

$30\%\text{ of valid votes}=2142$

$0.30\times 0.816x=2142$

$0.2448x=2142$

$x=\frac{2142}{0.2448}$

$x=8750$

Conclusion:
The total number of registered voters is 8750.